Finite Group Databasebeta

← #36351714

Subgroups — #36351714

Order 22144.

112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011286C211yesmaximal, characteristic, complemented
2#26011231C211yesmaximal, characteristic, complemented
3#454×C173C211yesmaximal, characteristic
4#19149594—21no—
5#19149590—21no—
62−1+4×C173C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C173—21no—
8#124×C173C2211yescharacteristic
9#104×C173—21no—
10#101×C173C411yescharacteristic, complemented
11C2×#13063436—41no—
12C22⋊Dic173—41no—
13#41×C173—21no—
14#41×C173—41no—
15#41×C173—41no—
16C2×Q8×C173—41no—
17C2×Q8×C173C2×C411yescharacteristic
18C2×D4×C173D411yescharacteristic
19SD16×C173—41no—
20D8×C173—41no—
21M16×C173—21no—
22C22⋊C4×C173—41no—
23C42×C173D411yescharacteristic
24C2×Dic173—81no—
25#13063436—42no—
26C23×C173—21no—
27Q8×C173—21no—
28Q8×C173—41no—
29Q8×C173—41no—
30D4×C173—21no—
31D4×C173—41no—
32D4×C173—41no—
33D4×C173—41no—
34C2×C4×C173—21no—
35C2×C4×C173—21no—
36C2×C4×C173—41no—
37C2×C4×C173—42no—
38C2×C4×C173C22⋊C411yescharacteristic
39C8×C173—41no—
40C22×C173—41no—
41C22×C173—41no—
42C22×C173—81no—
43C22×C173#9911yescharacteristic
44C4×C173—21no—
45C4×C173—41no—
46C4×C173—41no—
47C4×C173—41no—
48Dic173—161no—
49C2×C173—21no—
50C2×C173—81no—
51C2×C173—81no—
52C2×C173AΣL(1,𝔽16)11yescharacteristic
53C173#140411yescharacteristic, complemented, Sylow
54#1404—1731nomaximal, Sylow
55#454D17311yescharacteristic
56#354—1731no—
57#329—1731no—
582−1+4Dic17311yescharacteristic, complemented
59AGL(1,ℤ/8)—21no—
60#124C2×D17311yescharacteristic
61#104—21no—
62#101Dic17311yescharacteristic, complemented
63#99—3461no—
64#97—3461no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×Dic17311yescharacteristic
70C2×D4C22⋊D17311yescharacteristic
71SD16—41no—
72D8—41no—
73M16—21no—
74C2×C8—6921no—
75C22⋊C4—41no—
76C22⋊C4—6921no—
77C42C22⋊D17311yescharacteristic
78C23—21no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—41no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—41no—
89C2×C4—42no—
90C2×C4—13841no—
91C2×C4C22⋊Dic17311yescharacteristic
92C8—41no—
93C8—6922no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22#1914959411yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—27681no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#2601127811yescharacteristic
subgroupG/H
centerC2#26011278
commutatorC2×Q8×C173C2×C4
Frattini#124C2×D173
Fitting, prime radical#454×C173C2
socleC2×C173AΣL(1,𝔽16)
cosocle#124×C173C22
hypercenter#454D173
hypocenterC173#1404
LCSG ▷ C2×Q8×C173 ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #454

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26011286C2semidirect
2. #26011231C2semidirect
6. 2−1+4×C173C4semidirect
10. #101×C173C4semidirect
53. C173#1404semidirect
58. 2−1+4Dic173semidirect
62. #101Dic173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260112862
#260112312
#454×C1732
#1404173